Actual philosophy

That is, the universe is mathematical. Likewise, 'numbers' are themselves a universal construct. (Aliens will have numbers too, that work just like ours.)

Mathematical simply means relates to mathematics. This is obviously true but doesn't mean anything about where mathematics "comes from", or how it sits in the universe metaphysically.

I guess I'm arguing that our perception of them is a human construct. But they exist without us.

Perception is a human thing sure. And yes things exist without us.

Mathematics is knowledge (of things as they are). It is hubris to believe we need to comprehend that knowledge for it to exist -- which would make for circular reasoning indeed.

I think you are twisting the definition of knowledge as this is a trivially false statement under the usual use of the word. If we clarify your argument by replacing "knowledge" with "facts" ( if a tree falls but nobody hears it, did it still fall ? ). Then you are just repeating my argument yet conveniently redefining "math" in it's purest form as "facts". This is something I suggested you might do, but I am not willing to go so far as to posit this as a true statement.

Also consider that there are without a doubt facts that exist only in our minds and our constructs.

I think this is an issue of semantics, but I am willing to imagine this a calculation.

You're simply moving the question, not answering it. "The particles behave the way they do because of the nature of existence, not mathematics." Okay, so is the "nature of existence" mathematical? If not, then what is it, and why does it make things behave as if it was?

I think the nature of existence is mathematical, but that this is a much simpler and more trivial concept than many others would argue.

Maybe what I ultimately think is that the mathematics that humans do is a very different thing than the mathematics that the universe does. I think a big part of the problem is that we like to use one word to mean a bunch of different things.

Does the universe have an infinite amount of matter? If so, then every possible combination of matter is also exhausted infinitum. As in, there are an infinite amount of mes typing this same thing, some of them with better spelling.

It can follow that the universe needs no rules, no calculation, we're brute force. Some worlds have 2 plus 2 equal five, some might have 2 plus 2 equal 4 and 5 at the same time.

Or maybe I've just reiterated the original question in my own words: Must there be common rule(s) across all permutations of matter?

(If you want more on the whole "do we exist" thing, read some of Descartes's meditations. They're great stuff)

So, for this subject, I would suppose that we do, in fact, exist. This is just for the sake of argument here- otherwise this all just turns into dribble.

As for mathematics, you have to realize that whether humankind is aware or unaware of a particular theorem or axiom doesn't necessarily imply its nonexistence- this is why they are mathematical "discoveries," not mathematical "inventions." However, the issue I see here is that one has to see mathematics as a set of rules that, upon expansion, result in a plethora of structures and formations. We are not necessarily aware of all of these rules, but they still exist. At the same time, these rules create the foundation for which physics also abides by, not out of coincidence but out of nature. In a sense, nature is composed of mathematics- it is not a human construction, but the device of which constructed humankind.

in this context, from Dirac, Einstein, and many other highly distinguished religious physicists, come from.

Both Dirac and Einstein were not religious. Although they often have associated quotations involving the word 'God', it is misnomer that they were in fact themselves religious. Einstein believed simply in a deterministic universe and Dirac believed solely in a mathematical universe.

I thought today that the universe is always made of powerful mathematics that does brutal looking problems trivially. I think that all mathematics humans have dreamed up that do not describe something physical (mathematics outside of physics basically) are basically more inefficient, clumsy or incorrect ways to describe the processes that are going on. i.e. The universe (at it's most fundamental level) is using (is forced to use?) only the most efficient mathematical algorithms, whereas humans can be as inefficient and stupid as they like.

behaves exactly like modulo, yet it has nothing to do with division. Is it inappropriate to treat it as modulo in a proof?
Given "only" infinitely many bits and a computer capable of completing infinite programs in finite time, you can implement any function to and from Rn. Can we talk about such a program as if it was the actual function?

Fun fact: An infinite bit string is a real. The odd and the even bits of an infinite bits string are two reals. The ith bits, for all i congruent to n modulo m, is an array of n reals. All the ith bits, for i divisible by the nth prime, is an infinite array of reals. And then we can start talking about primeth primes and such.

Technically, modulo is defined declaratively in terms of an expression of the form a = nq + r.
What I mean is that no actual division or multiplication is performed. The equality of that function to modulo might even be considered an interesting side effect.

I fear this discussion is going to veer towards what it means to "perform" an operation. Can we really say that something was divided when we don't have the result at the end? Yes, the algorithm is one counter away from being proper division, so does that make it division?

Well, I think the discussion is headed a bit away from the original premise, so let me try and steer it back.

For most of this discussion, the numbers that we are referring to are those on the traditional real number line, whose distance is a result of the subtraction of any two points. But what if we changed it? What if the distance between 5 and 7 was not the result of subtracting the two, but the result of a much more complicated equation? There is a particular number system known as the p-adic number system, where any two numbers are said to be closer together so long as they are divisible by a higher power of p^v, where p is a prime number. I'd go in depth, but I just spent what looks to be an hour struggling with the Wikipedia page only to lose yet another part of my sanity to the math pages of Wikipedia. However, my point still stands- if numbers can be redefined so easily, then what numbers are we even dealing with here?

My problem is not with straying away. Discussions on definitions of terms are just slightly less interesting than "who would win" arguments.

As I understand it, after two minutes of skimming, the p-adic number system is a different way of encoding numbers, not an actual redefinition of numbers.
I don't even know what it would mean to "redefine" a number. Wouldn't it just be a mapping of some sort?

From my point of view, Math is a language. It describes real things (patterns), but it isn't any more "real" then English is. Numbers and operations are just the words, the syntax. Just like the word "dog", whether it is the sound you make, or the letters spelled out, isn't an actual dog, the number 5, isn't anything, it describe something.